Showing posts with label School Mathematics from Higher Viewpoints. Show all posts
Showing posts with label School Mathematics from Higher Viewpoints. Show all posts

2024-11-17

4: Are You Talking About Sets or About Collections?

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Japanese schools talk about "shugo"s, but are they sets or collections?

Topics


About: elementary school mathematics

The table of contents of this article

Starting Context

Target Context


  • The reader will know the distinction between sets and collections.

Orientation


There is an article on becoming a benefactor of humanity by being a conduit of truths

There is an article on definition of set and Russel's "paradox".

Main Body

1: What Does "shugo" (Used in Japanese Schools) Mean?


Special-Student-7-Hypothesizer
 Japanese schools talk about "shugo" s.

For example, they think of the "shugo" of some 3 apples, the "shugo" of the students of a class, etc..

Special-Student-7-Rebutter
 What is "shugo" exactly?

Special-Student-7-Hypothesizer
 Any "shugo" is any collection of some objects such that the membership is unambiguous.

Special-Student-7-Rebutter
 Is there ever a collection such that the membership is ambiguous?

Special-Student-7-Hypothesizer
 I do not think there is: the concept of 'collection' itself includes the requirement that the membership is unambiguous; the qualification is just for emphasis.

Special-Student-7-Rebutter
 So, is "shugo" nothing but 'collection' in English?

Special-Student-7-Hypothesizer
 I can just guess so, because Japanese textbooks do not say what "shugo" is called in English, but dictionaries usually say that "shugo" in mathematics is "set" in English.

Special-Student-7-Rebutter
 But 'set' and 'collection' are 2 different things in mathematics.

2: Caution: There Are Some Multiple Set Theories


Special-Student-7-Hypothesizer
 As a caution, there are some multiple set theories, and what 'set' is in each theory depends on the theory.

Special-Student-7-Rebutter
 Very confusing; cannot the theories use different names?

Special-Student-7-Hypothesizer
 It is very confusing, but as each theory exists because it regards the other theories unsatisfactory, each theory says "'set' we call is the real 'set'!", and each theory uses "set".

Special-Student-7-Rebutter
 I kind of understand, but ...

Special-Student-7-Hypothesizer
 Anyway, arguably the most popular one is the ZFC set theory, and we will mean the ZFC set theory by "the set theory" and mean 'set' in the ZFC set theory by "set" hereafter.

Note that what we are going to say here are not particularly canonical; as we have not seen any convincing argument, we are presenting a hypothesis that is convincing for us. Nobody should accept something just because someone (whoever he or she is) says so or just because many people say so; check yourself whether the hypothesis is unboundedly consistent, which is the only way to approach truths.

3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?


Special-Student-7-Hypothesizer
 Most textbooks cite Russel's "paradox" as the reason why the set theory needs to be as it is, but the arguments are not convincing at all, at least for us.

Special-Student-7-Rebutter
 What are the arguments like?

Special-Student-7-Hypothesizer
 Historically, there was the so-called "naive set theory", which is based on the so-called "naive comprehension axiom", which is "Any precisely specified property of member can be used to define a set.".

Then, Russel's "paradox" came along and refuted the "naive comprehension axiom", and so refuted the "naive set theory".

So, now, the ZFC set theory admits only the empty set and the things constructed from the empty set as "set".

So, the collection of some 2 electrons are not admitted to be "set", because the collection is not constructed from the empty set.

Special-Student-7-Rebutter
 There seems a wide gap in the reasoning: why would the collection of the 2 electrons not be "set" just because the "naive comprehension axiom" has been refuted?

Special-Student-7-Hypothesizer
 I have not seen any convincing explanation.

In fact, why will we not have the modified axiom that "Any precisely specified property of member that (the property) determines membership unambiguously can be used to define a set."?

Special-Student-7-Rebutter
 What Russel's "paradox" says is that just "precisely specified property of member" does not guarantee the unambiguous-ness of membership, so, why will we not add the unambiguous-ness as an additional requirement?

Special-Student-7-Hypothesizer
 I have not seen any convincing argument why; it seems just that as checking it is not easy in general, they want a way to be able to define sets without bothering to do checking.

Special-Student-7-Rebutter
 Well, is that an attitude to humbly approach truths? That seems an attitude to ignore whatever are inconvenient for humans.

Special-Student-7-Hypothesizer
 Whether the unambiguous-ness can be checked easily is just a matter of human convenience, and the essence of the ZFC set theory is that it gave up covering the concept of "collection" and has decided to degenerate to deal with only convenient-for-humans kind of collections, which are now called "sets", which is my understanding.

In fact, I do not say at all that having a theory about such limited kind of collections is bad, but we need to be aware what the theory is really doing.

Special-Student-7-Rebutter
 We need to avoid some confusions: "naive set theory" means the theory with the "naive comprehension axiom" not the concept of 'collection'; while the ZFC set theory excludes most collections like a collection of some 2 electrons, that is not because the collection of 2 electrons are inappropriate but because the set theory has degenerated.

Special-Student-7-Hypothesizer
 The main cause of confusions is that the ZFC set theory keeps using the term, "set", while the concept of 'set' has degenerated from equaling 'collection' to 'convenient-for-humans collection'.

4: In Fact, a Collection in General Is Not Defined by Any Property of Member


Special-Student-7-Hypothesizer
 In fact, I think that the "naive comprehension axiom" is more fundamentally problematic than Russel's "paradox".

The problem is that it tries to define any collection by a property of member.

For example, let us think of the collection of some balls I put into a bag.

I put a ball into the bag just because my groping hand happened to touch the ball, not because the ball is red or blue or something.

Special-Student-7-Rebutter
 You mean that the collection is not defined by any property of ball, like being red or something?

Special-Student-7-Hypothesizer
 Yes.

Special-Student-7-Rebutter
 Your collection is usually defined by enumerating the balls, as the collection is finite: you cannot put infinitely many balls into the bag by groping.

Special-Student-7-Hypothesizer
 Yes, when the collection is finite, there is the escape.

But what if the collection is infinite?

Special-Student-7-Rebutter
 How will you define an infinite collection not by property of member?

Special-Student-7-Hypothesizer
 If we ignore Relativity, let us take a \(\mathbb{R}^3\) Cartesian coordinates system for the Universe, and take the 1-radius open ball around each rational-coordinates point. Then, the numbers of the electrons in each open ball at a time, which is a subcollection of the natural numbers set, is my infinite collection.

Special-Student-7-Rebutter
 You mean that the collection is not defined by any property of natural number, like being even or something?

Special-Student-7-Hypothesizer
 Yes.

Special-Student-7-Rebutter
 It does not seem to be really infinite, supposing that there are only some finite number of electrons in the Universe.

Special-Student-7-Hypothesizer
 Then, let us take the pair of the center-coordinates and the number of the electrons, like ((1.2, 3.4, 5.6), 7).

Special-Student-7-Rebutter
 Then, certainly, the collection is infinite.

What if we are not allowed to ignore Relativity?

Special-Student-7-Hypothesizer
 Let us take a chart for the spacetime manifold, whose (the chart's) domain is homeomorphic to \(\mathbb{R}^4\), and let us do the same thing for that \(\mathbb{R}^4\).

Special-Student-7-Rebutter
 So, you are saying that such collections exist in the reality but the "naive comprehension axiom" cannot grasp such collections.

Special-Student-7-Hypothesizer
 In fact, that is also the problem of the "restricted comprehension axiom", which is in fact the same with the "naive comprehension axiom" except that the "restricted comprehension axiom" thinks of only the elements of an already-known-to-be-set collection.

Special-Student-7-Rebutter
 Is "the number of the electrons in the open ball" not 'property of member'?

Special-Student-7-Hypothesizer
 At least, the ZFC set theory does not admit such a property: the "restricted comprehension axiom" requires that the property is expressed as a formula that allows only the specified operators like \(\in\) and some already-known-to-be-set collections.

Special-Student-7-Rebutter
 Did the "naive comprehension axiom" admit such a property?

Special-Student-7-Hypothesizer
 Maybe, but such a usage of the term, "property of member", would be quite harmful (a too-far-fetched interpretation, I would say), in my opinion. Being "the number of the electrons in the open ball" is not any property of the natural number but is a property of the Universe: I mean, being "the number of the electrons in the open ball" is not about the natures of the natural number (like being even, prime, or something) but about the natures of the Universe.

Special-Student-7-Rebutter
 Anyway, the "restricted comprehension axiom" does not allow your collection.

Special-Student-7-Hypothesizer
 So, there is the collection in the reality, but the ZFC set theory refuses to cope with the collection.

5: "shugo" Means 'Collection'


Special-Student-7-Hypothesizer
 So, "shugo" in Japanese textbooks means 'collection' not "set".

And when a Japanese student later hears of Russel's "paradox" and hears '"set" of some 2 electrons' refuted, the "shugo" of the 2 electrons is not refuted at all, because the collection of the 2 electrons is not refuted at all.

And when he or she later hears that "naive set theory" is inappropriate, the concept of "shugo" (the concept of 'collection') is not inappropriate at all, because the concept of 'collection' is not based on the naive comprehension axiom.

6: The Concept of 'Collection' Lives on


Special-Student-7-Hypothesizer
 The concept of 'collection' is still valid and is indeed used in mathematics.

As a typical example, in the category theory, a category in general is not any set but a collection. For example, the category of all the sets, \(Set\), is a collection but not any set.

Special-Student-7-Rebutter
 \(Set\) is prevalently called "class".

Special-Student-7-Hypothesizer
 Yes, 'class' is a concept wider than 'set' and narrower than 'collection', but anyway, a class is not any set in general.

Special-Student-7-Rebutter
 How is 'class' narrower than 'collection'?

Special-Student-7-Hypothesizer
 Roughly speaking, any class is still constructed from the empty set and is still defined by property of member with a formula.

Anyway, if something was inappropriate just because it was not "set" in the ZFC set theory, the whole category theory would be inappropriate.

And "collection" is still frequently used in many mathematical textbooks: while a reader may wonder why "collection" is being used instead of "set", that is because the mentioned object is not or is not necessarily any "set" according to the ZFC set theory, and if 'collection' was inappropriate, such textbooks would be inappropriate, which is not the case.

7: Mathematics Is Not Exactly Built on the Set Theory


Special-Student-7-Hypothesizer
 A rather prevalent misconception is that "whole mathematics is built on the set theory.".

Special-Student-7-Rebutter
 That is what the naive set theory aspired to do.

Special-Student-7-Hypothesizer
 But the attempt failed (by mainly Russel's "paradox") and because the set theory gave up dealing with general collections and has degenerated to be a theory on a very limited kind of collections, the aspiration is given up.

Certainly, the set theory is still a very important part of mathematics, but it is not exactly the basis of whole mathematics.

Special-Student-7-Rebutter
 A proof is that "collection" is still used in many mathematical textbooks.

References
<The previous article in this series | The table of contents of this series |

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2024-07-01

3: What Are Discrete? What Are Continuous?

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Water is not continuous, because it is made of molecules. Even if we ignore the fact, "Water is continuous." is mathematically nonsense.

Topics


About: elementary school mathematics

The table of contents of this article

Starting Context

Target Context


  • The reader will know what 'discrete' or 'continuous' means in mathematics.

Orientation


There is an article on becoming a benefactor of humanity by being a conduit of truths

Main Body

1: Water Is Not Continuous


Stage Direction
Special-Student-7-Hypothesizer has a paperback in his hands, turning over some pages back and forth.

Special-Student-7-Hypothesizer
 This book, an enlightening book on mathematics (a good one in that it arouses interests in mathematics, whatever we say about it hereafter), says that there are discrete amounts and continuous amounts, and cites water as an example of continuous amounts.

Special-Student-7-Rebutter
 I do not understand a bit: while any chunk of water is a set of molecules, how is water a "continuous amount"?

Special-Student-7-Hypothesizer
 Obviously, the author ignored the fact that water was made of molecules.

Special-Student-7-Rebutter
 Then, what water was supposed to be?

Special-Student-7-Hypothesizer
 Water seems to be have been supposed to be what the ancients imagined to be.

Special-Student-7-Rebutter
 What did the ancients imagine?

Special-Student-7-Hypothesizer
 Well, ... I do not know how to state it.

Special-Student-7-Rebutter
 But modern mathematics says that the chunk of water is made of some points anyway, even if the points are not suppose to be the molecules, does it not?

Special-Student-7-Hypothesizer
 I think so: the only way of describing the chunk of water by modern mathematics is to regard the chunk of water as a set of some points, whether the set is a set of finite number of molecules or a set of infinite number of points, as far as I know.

Special-Student-7-Rebutter
 And how is the set of infinite number of points "continuous amount"?

In the 1st place, what does "amount" mean? Is the set itself "amount", is the number of points "amount", is the volume of the chunk "amount", is the mass of the chunk "amount", or what?

Special-Student-7-Hypothesizer
 The book seems to be saying that the chunk of water itself is a "continuous amount".

Special-Student-7-Rebutter
 I do not understand a bit.

Special-Student-7-Hypothesizer
 The book compares the chunk of water with a set of apples and says that water is continuous because while each element of the apples set is separated and independent from the other elements, that is not the case for the chunk of water.

Special-Student-7-Rebutter
 I disagree. Each point of water is separated from any other point: for example, denoting any point as \((x, y, z)\), any other point will be \((x', y', z')\) such that \((x, y, z) \neq (x', y', z')\), and while the distance of the 2 points, \(\sqrt{(x' - x)^2 + (y' - y)^2 + (z' - z)^2}\), is positive, how are the 2 points not separated? Besides, what does "independent" mean? Any distinct 2 points of any set should be independent, because they are different points: otherwise, they would be the same point.

Special-Student-7-Hypothesizer
 What the author wanted to say seems that between any distinct points of water, there is another point of water.

Special-Student-7-Rebutter
 That is the case also for the rational numbers set (I am not talking about the real numbers set) with the canonical order, but mathematics does not call the rational numbers set "continuous", as far as I know.

Special-Student-7-Hypothesizer
 The book also says like "Water can be divided infinite times to still be water and when any 2 chunks of water are united, the union is seamless.".

Special-Student-7-Rebutter
 I sincerely disagree: any single point of water cannot be divided, while any infinite set of apples can be divided infinite times: for example, supposing that the apples are indexed with the natural numbers, divide the set into the even-indexed set and the rest; divide the even-indexed set into the 4-multiple-indexed set and the rest; and so on; seamless? but the distinction of whether a point is from the 1st chunk or from the 2nd chunk still exists, and I call the distinction "seam": it is just a matter of that we think of the imaginary partition board in the united water.

Special-Student-7-Hypothesizer
 Certainly, whether a set can be divided infinite times is just about whether the set is infinite; whether a set is seamless is just about whether the union is of sets with some common points: if there is no common point, the union has the seam defined by the origins of the points.

Special-Student-7-Rebutter
 Then, how is the chunk of water "continuous amount"?

Special-Student-7-Hypothesizer
 The book also says that the elements of the apples set can be counted, while the elements of the water chunk set cannot be counted.

Special-Student-7-Rebutter
 Huh? Are we still talking about continuousness? That was about the distinction between countable sets and uncountable sets.

Special-Student-7-Hypothesizer
 Mathematics distinguish countable sets and uncountable sets but does not call uncountable sets "continuous".

2: Continuousness Cannot Be Defined as a Property of Set


Special-Student-7-Hypothesizer
 Continuousness cannot be defined as a property of set, because any pure set is just a collection of elements without any relation between elements assumed.

Special-Student-7-Rebutter
 The word, "continuous", insinuates relations between elements, but the concept of 'set' has eliminated relations between elements.

Special-Student-7-Hypothesizer
 In mathematics, abstraction is crucial: once any concept has been defined, it is important that any undue property that has been eliminated from the concept is never surreptitiously assumed.

Special-Student-7-Rebutter
 The statement, "between any distinct points of water, there is another point of water", is not about any pure set, but about a set with an order, while the order is an extra structure put onto the set.

Special-Student-7-Hypothesizer
 To distinguish concepts clearly is the 1st step for clear thinking, and mathematics is arguably the best way for teaching making clear distinctions of concepts, but lower-level schools are missing the opportunity.

In fact, whether one is good with mathematics is not about being good with numbers (mathematics is not always about numbers), but about being good with making clear distinctions of concepts.

In fact, arguing with a mathematics-hater is very futile, because he or she incessantly confuses things and his or her arguments are just muddles.

Special-Student-7-Rebutter
 And there are so many mathematics-haters ...

Special-Student-7-Hypothesizer
 Continuousness is not about set or about set with order: certainly, we can think of whether another element exists between any 2 distinct elements in a set with an order, but mathematics does not call that property "continuousness".

Special-Student-7-Rebutter
 I understand that all cannot be taught in lower-level schools, but I claim that no lie should be taught in whatever level.

Special-Student-7-Hypothesizer
 Not teaching something is OK, but teaching a lie is not OK, while "Water is continuous." is a lie.

Special-Student-7-Rebutter
 Such lies may be "easy" for many unenthusiastic students, but sincere students will be confused by such lies, because lies do not make sense if sincere students contemplate them.

Special-Student-7-Hypothesizer
 Education on the Bias Planet is giving priority to pampering unenthusiastic students over not-confusing sincere students, because unenthusiastic students are many while sincere students are very few: it is called "democracy".

3: Continuousness is About Measure


Special-Student-7-Hypothesizer
 In fact, continuousness is about measure.

Special-Student-7-Rebutter
 What is 'measure'?

Special-Student-7-Hypothesizer
 Any measure is an extra structure put onto a set that (the extra structure) is defined like this: we think of a \(\sigma\)-algebra, which is a subset of the power set (the set of all the subsets of the set) of the set, and map to each element of the \(\sigma\)-algebra an element of \([0, \infty]\), while the \(\sigma\)-algebra and the map have to satisfy certain conditions.

That may sound enigmatic to many students, but for example, the method of taking volumes of water chunks is a measure.

The reason for choosing a \(\sigma\)-algebra instead of using the power set itself is that we do not need to measure all the subsets (so, we call each element of the \(\sigma\)-algebra "measurable subset").

Special-Student-7-Rebutter
 That is the reason why I asked what "amount" meant: "Water is continuous." is nonsense because any measure is not specified.

Special-Student-7-Hypothesizer
 Note that we can think of many measures for the same set: we can take a measure that takes volumes, a measure that takes masses, the measure that counts numbers of points (called "counting measure"), and so on; if we choose different units of volumes or masses, the measures are different.

Special-Student-7-Rebutter
 Can we take the counting measure for the chunk of water while the set is uncountable?

Special-Student-7-Hypothesizer
 Yes we can. The counting measure maps to each infinite measurable subset \(\infty\) and maps to each finite measurable subset the number of the points, so, the set's being uncountable does not prevent the counting measure from being valid.

Special-Student-7-Rebutter
 So, although the book is as though the chunk of water cannot use the counting measure, that is not true.

Special-Student-7-Hypothesizer
 That is certainly not true, although most people do not use the counting measure for daily purposes.

Special-Student-7-Rebutter
 And how is a measure 'continuous'?

Special-Student-7-Hypothesizer
 Any measure is continuous if and only if each single point subset measures 0.

Special-Student-7-Rebutter
 That is rather unexpectedly simple.

Special-Student-7-Hypothesizer
 The volume measure for water is indeed continuous, because the volume of each single point subset is 0.

On the other hand, the counting measure for the chunk of water or the apples set is not continuous, because each single point subset measures 1.

Special-Student-7-Rebutter
 So, "Water is continuous." is indeed a nonsense, because the chunk of water can have a non-continuous measure.

Special-Student-7-Hypothesizer
 As continuousness is about measure, we need to specify what measure we have chosen.

What the book should have said is "For water, humans usually choose continuous measures, practically speaking.", while the reasons why such a measure is continuous the book cites (like "points are separated"; "infinitely divisible"; "elements are uncountable") are all wrong.

4: What Is Discrete Measure?


Special-Student-7-Rebutter
 What is discrete measure? Is every non-continuous measure discrete?

Special-Student-7-Hypothesizer
 No.

'Discrete measure' is defined to be any measure such that there is a countable measurable subset whose complement (which is guaranteed to be a measurable subset by the definition of \(\sigma \)-algebra) measures 0.

Special-Student-7-Rebutter
 In the case of the counting measure for the apples set, ...

Special-Student-7-Hypothesizer
 As the whole set is countable, the "countable measurable subset" can be taken to be the whole set, and the complement is the empty set, which measures 0. So, the counting measure for the apples set is discrete.

Special-Student-7-Rebutter
 So, the counting measure for each countable set is discrete.

How about the counting measure for the chunk of water?

Special-Student-7-Hypothesizer
 There is no such a countable subset, so, is not discrete.

Special-Student-7-Rebutter
 So, the counting measure for the chunk of water is not continuous or discrete.

Special-Student-7-Hypothesizer
 Also the volume measure for the chunk of water is not discrete.

5: Continuousness Is Not About Covering a Line


Special-Student-7-Rebutter
 What the word, "continuous", reminds most people of may be 'covering a line'.

Special-Student-7-Hypothesizer
 Ah, but mathematics does not call it "continuous", as far as I know.

And note that "divisible infinite times" or "having a point between any 2 points" does not imply 'covering a line'.

In fact, the rational numbers set on a line is 'divisible infinite times' and 'has a point between any 2 points', but it does not cover the line.

And note that the real numbers set is just a set and the points do not have inherent locations: if you imagine that the real numbers set is on a line, that is just because you are imagining so: you can also imagine the real numbers scattered around the universe.

Special-Student-7-Rebutter
 When you talk about the real numbers set on a line, you are talking about the Euclidean metric space, not about the pure real numbers set.

Special-Student-7-Hypothesizer
 Mathematics talks about 'connected metric (or topological) space' but not about "continuous metric (or topological) space", as far as I know.

Any connected topological space (any metric space is canonically a topological space) is defined to be a topological space that is not the union of any disjoint nonempty open subsets, which may not be understood by people who have not learned topology, but you should at least know that criteria like "divisible infinite times" do not define 'connectedness'.

Of course, 'covering a line' is an important issue that historically prompted real numbers, but that is different from being continuous.

6: Continuousness May Be About Map


Special-Student-7-Rebutter
 There is also the concept of 'continuous map', right?

Special-Student-7-Hypothesizer
 Yes, we have talked about continuous measure so far because the book should have meant it, but "continuous" is used in some other cases.

"continuous map" is arguably the most famous one of them.

Special-Student-7-Rebutter
 What does that mean?

Special-Student-7-Hypothesizer
 Continuousness is in fact talked about each point of the domain of the map, and means that for each neighborhood of the value point on the codomain, a neighborhood of the domain point is mapped into the codomain neighborhood (the knowledge of 'topology' is required in order to understand that). Continuous map means that the map is continuous at each domain point.

Special-Student-7-Rebutter
 How about "discrete map"?

Special-Student-7-Hypothesizer
 I have not heard that concept.

7: What Should Be Taught to Low-Level Schools Students


Special-Student-7-Rebutter
 But what should be taught to low-level schools students? I mean, should they be taught the exact conditions for \(\sigma\)-algebra or measure?

Special-Student-7-Hypothesizer
 I do not particularly say so.

In fact, the concept, 'continuous', does not need to be introduced to them.

Special-Student-7-Rebutter
 Does it not?

Special-Student-7-Hypothesizer
 The real purpose is just to convince them that natural numbers are not enough; they need fractions and decimals. So, just convince them so, without introducing 'continuousness'.

Special-Student-7-Rebutter
 Certainly, 'continuousness' is not indispensable for introducing fractions and decimals.

Special-Student-7-Hypothesizer
 As an example, line segment length is better than water volume, because water really consists of molecules, which complicates the situation.

Special-Student-7-Rebutter
 But students need to know about water volume anyway.

Special-Student-7-Hypothesizer
 Students definitely need to know about approximations. While water really consists of molecules, we make an approximation when we measure the volume of any chunk of water.

Special-Student-7-Rebutter
 Yes, 'approximation' and 'error' may be the most important concepts that even low-level schools students need to learn.

Special-Student-7-Hypothesizer
 While any chunk of water is made of some molecules and the molecules are moving around, what is indeed the volume of the chunk of water? ... However we exactly define 'volume', the volume is really wobbling, and your measured volume has an error, partly because the volume is wobbling and partly because your act of measuring is inaccurate.

Special-Student-7-Rebutter
 It is crucial to understand that that does not mean that the reality itself is ambiguous: you cannot say that the reality is ambiguous just because you cannot measure the reality accurately.

Special-Student-7-Hypothesizer
 Unfortunately, that is what is not understood even by many renowned scientists.

Special-Student-7-Rebutter
 What should be taught about 'set'?

Special-Student-7-Hypothesizer
 Any set is just a collection of some elements with no extra property assumed, and when you measure a set, you need to define an extra structure, which even low-level schools students need to learn.

Special-Student-7-Rebutter
 At least, the students need to understand that the chunk of water does not automatically dictate any measure: there can be a volume measure, a mass measure, the counting measure, and a fancy measure someone can concoct, and they need to specify the measure they are going to talk about.

Special-Student-7-Hypothesizer
 That is really a matter of of course: without our choosing how to measure the chunk of water, the value of the measuring does not appear, but that knowledge seems to be supposed to be difficult to be understood.

Special-Student-7-Rebutter
 As there can be multiple measures for a set, the concept, 'set', has to be established independent of any measure. Without understanding that way of thinking, mathematics cannot be really understood.

Special-Student-7-Hypothesizer
 In fact, that way of thinking is the thing that students most need to learn, but they are being pampered on the excuse, "They won't understand anyway.".

References
<The previous article in this series | The table of contents of this series | The next article in this series>
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2023-07-02

2: Why Are Mirror Images Regarded to Be Congruent in Plane Geometry?

<The previous article in this series | The table of contents of this series | The next article in this series>
Just a definition by a mathematician? But there should be some good reasons for adopting the definition.

Topics


About: junior high school mathematics

The table of contents of this article

Starting Context

Target Context


  • The reader will know some reasons why mirror images are regarded to be congruent in the plane geometry.

Orientation


There is an article on becoming a benefactor of humanity by being a conduit of truths

Main Body

1: Why Are Mirror Images Regarded to Be Congruent in the Plane Geometry?


Special-Student-7-Rebutter
 In the (2-dimensional) plane geometry, 2 figures one of which is a mirror image of the other seems to be regarded to be congruent, at least in the Japanese junior high school mathematics.


Special-Student-7-Hypothesizer
 The Japanese junior high school mathematics seems to be saying that 2 figures one of which can be "moved" to coincide with the other is regarded to be congruent.

Special-Student-7-Rebutter
 What does "move" exactly mean?

Special-Student-7-Hypothesizer
 It is the problem that "move" is not defined explicitly in the Japanese junior high school mathematics, although I understand that junior high school students in general are not expected to understand rigorous definitions.

Special-Student-7-Rebutter
 But saying "move" is meaningless if the term is not understood accurately.

Special-Student-7-Hypothesizer
 Intuitively speaking, a triangle is imagined like a physical object, which is moved in the space: as normal junior high school students cannot teleport objects, they will imagine continuously moving the object in the space, without deforming the object.


Special-Student-7-Rebutter
 I see 2 problems there.

1st, how can you be sure that the object is not being inadvertently deformed while you are moving the object?

Special-Student-7-Hypothesizer
 Well, the physical object is supposed to be very hard and be never deformed unintentionally by my just moving it.

Special-Student-7-Rebutter
 Is "very hard" a legitimate mathematical concept?

Special-Student-7-Hypothesizer
 "very hard" is a physical assumption that if the object is wooden or something, the object will not be deformed, at least much, by being moved without being intentionally tortured. ... I know that it is not a refined mathematical concept, but it seems to have been a historical motivation for thinking of 'congruence'.

Special-Student-7-Rebutter
 2nd, what does "the space" mean when you say "is moved in the space"? As we are talking about a plane, "the space" should be naturally construed to be the plane, I think. Then, a figure cannot be "moved" to coincide with its mirror image by moving the figure continuously in "the space".

Special-Student-7-Hypothesizer
 Apparently, the Japanese junior high school mathematics "moves" the figure in the ambient 3-dimensional space.


Special-Student-7-Rebutter
 I do not say that they cannot introduce the ambient 3-dimensional space, but it seems unwise: while we should have been able to concentrate on the 2-dimensional plane, why do they have to complicate the situation by introducing the extra dimension?.

Special-Student-7-Hypothesizer
 They seem to have cursorily introduced the ambient 3-dimensional space, because they are familiar with the 3-dimensional space. However, they will have to introduce the ambient 4-dimensional space for the 3-dimensional space geometry, if they are consistent with the strategy.

Special-Student-7-Rebutter
 Certainly, 3-dimensional figures can be "moved" in the ambient 4-dimensional space, but it seems to be against the purpose of being intuitive: why do they have to make students think of the hard-to-imagine 4-dimensional space while we are concerned only with the 3-dimensional space?

Special-Student-7-Hypothesizer
 Introducing the ambient space seems to be a strategy that is natural only for the plane geometry, and employing a strategy that lacks generality seems unwise.

2: It Seems More Natural to Not Call Mirror Images 'Congruent', Supposing That 'Congruent' Means Being Same-Shaped and Same-Sized


Special-Student-7-Hypothesizer
 In fact, supposing that 'congruent' means being same-shaped-and-same-sized, calling mirror figures 'same-shaped' seems unnatural, in the first place.

Special-Student-7-Rebutter
 Are there some grounds for that supposition?

Special-Student-7-Hypothesizer
 At least most general (not mathematical) dictionaries say that 'congruent' means being same-shaped-and-same-sized.

Special-Student-7-Rebutter
 Mirror images are not same-shaped in my vocabulary.

Special-Student-7-Hypothesizer
 If someone feels mirror images being same-shaped, the reason seems to be that he or she has imagined moving figures in the ambient higher-dimensional space.

Special-Student-7-Rebutter
 But most people do not naturally imagine moving 3-dimensional figures in the ambient 4-dimensional space, I guess.

3: Some Reasons Why We Should Think of Map Instead of "Move"


Special-Student-7-Hypothesizer
 In fact, introducing "move" seems not wise, while "move" is understood as 'continuously move', as is naturally so.

A reason is that a figure has to be taken out of the plane into the ambient 3-dimensional space, if mirroring wants to be regarded to be a kind of "move".

Special-Student-7-Rebutter
 As has been discussed above.

Special-Student-7-Hypothesizer
 Another reason is that we have to think of the entire continuous movement of the figure in order for us to talk in terms of "move", while we are interested only in the original figure and the terminal figure.

So, why do we not skip the unnecessary middle?

Special-Student-7-Rebutter
 If there is a reason, it seems to be that they are not freed from the intuitive mental picture of moving a physical object.

Special-Student-7-Hypothesizer
 But in order for us to guarantee that the figure is not deformed in being moved, we have to guarantee that the figure is not deformed at each position, and then, why do we not guarantee that the figure is not deformed just at the terminal position?

Special-Student-7-Rebutter
 I do not guess that there is a legitimate reason.

Special-Student-7-Hypothesizer
 So, let us think of 'map' instead of "move", where 'map' means mapping each point of the original figure to a point of the terminal figure, without bothering to move the point continuously in the space.

The merits are the reverse of the unwiseness of introducing "move": we do not need the ambient space and we do not need to think of the middle.

Any map is prevalently denoted like \(f: S_1 \rightarrow S_2\), where \(S_1\) and \(S_2\) and some sets, and \(S_1\) is called 'the domain of the map' and \(S_2\) is called 'the codomain of the map'. The map does not need to really map the domain to the whole codomain, and the really mapped part of the codomain is called 'the range of the map'; if the range equals the codomain, the map is called to be a surjection. If any different 2 elements of \(S_1\) are mapped to different elements, the map is called to be an injection. Any map that is a surjection and an injection is called a bijection.

4: 2 Figures Are Congruent iff 1 of Them Is the Image of the Other Under an Isometry


Special-Student-7-Hypothesizer
 In fact, mathematically speaking, 2 figures are congruent iff 1 of them is the image of the other under an isometry, where any isometry is any map that preserves the distance between any 2 points.

In the plane geometry, with the sets of 2 figures denoted as \(S'_1 \subseteq \mathbb{R}^2\) and \(S'_2 \subseteq \mathbb{R}^2\), \(S'_1\) and \(S'_2\) are congruent if and only if there is an isometry, \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), such that \(f (S'_1) = S'_2\).

Special-Student-7-Rebutter
 Are mirror images congruent by the definition?

Special-Student-7-Hypothesizer
 Yes: the lengths of any corresponding sides of any 2 mirror triangles are the same.

Note that "preserves the distance between any 2 points" means that with the distance between 2 points, \(p_1, p_2\), denoted as \(dist (p_1, p_2)\) and the image of \(p_i\) denoted as \(f (p_i)\), \(dist (f (p_1), f (p_2)) = dist (p_1, p_2)\).

Special-Student-7-Rebutter
 Is any angle automatically preserved?

Special-Student-7-Hypothesizer
 Yes. Although the definition of isometry talks only about lengths, any triangle, \(ABC\), is mapped to \(A'B'C'\), and the angle, \(\angle ABC\), equals the angle, \(\angle A'B'C'\), because the side length, \(CA\), equals the side length, \(C'A'\), in addition that \(AB = A'B'\) and \(BC = B'C'\).

5: Isometry Is Really Any Combination of Translation, Rotation, and Mirroring


Special-Student-7-Hypothesizer
 Any isometry is really any combination of translation, rotation, and mirroring.




Special-Student-7-Rebutter
 Well, is that obvious?

Special-Student-7-Hypothesizer
 Although we do not show any rigorous proof, it is intuitively obvious.

Any translation preserves distances.

Any rotation preserves distances.

Any mirroring preserves distances.

And any isometry has to be a combination of translation, rotation, and mirroring, because when a triangle, \(ABC\), is mapped to \(A'B'C'\) by the isometry, \(A'B'C'\) is same-shaped-and-same-sized with or a mirror image of \(ABC\) (what else can it be indeed?), and \(A'B'C'\) has to have been mapped from \(ABC\) by a combination of translation, rotation, and mirroring.

Special-Student-7-Rebutter
 We are supposing that if \(A'B'C'\) is same-shaped-and-same-sized, it has to be a combination of translation and rotation, and if \(A'B'C'\) is a mirror image, it has to be a combination of translation, rotation, and mirroring.

Special-Student-7-Hypothesizer
 Although we have not proved the supposition rigorously, it seems intuitively obvious.

6: But Anyway, Why Is Isometry Used in the Definition of 'Congruence'?


Special-Student-7-Rebutter
 I understand the definition of 'congruence' with isometry, but why "isometry"?

Special-Student-7-Hypothesizer
 That is the point on which we have begun this article. Is a definition, "2 figures are congruent iff 1 of them is the image of the other under a combination of translation and rotation.", not more natural?

Someone may say that it is just that a mathematician (in fact, Euclid, I guess) just adopted the definition with isometry, but I think that that mentality is not good: mathematics is not any discipline in which students have to blindly accept an unreasonable definition just because an authority made the definition; we can and should ask whether there are good reasons to adopt the definition.

Special-Student-7-Rebutter
 What are the good reasons for the definition of 'congruence' with isometry?

Special-Student-7-Hypothesizer
 Some typical concerns in the plane geometry is that the length of a line segment equals the length of another line segment and an angle equals another angle, and isometry is sufficient for addressing such concerns.

We are typically concerned with whether an angle \(\angle ABC\) equals another angle \(\angle A'B'C'\), and it does not really matter whether the triangle, \(A'B'C'\), is same-shaped-and-same-sized with or a mirror image of the triangle, \(ABC\).

Special-Student-7-Rebutter
 Although there may be some cases in which it matters, but it is more economical to distinguish between being same-shaped-and-same-sized and being a mirror image only in such rare cases.

Special-Student-7-Hypothesizer
 That is the good reason why we have the concept that is defined with isometry.

Special-Student-7-Rebutter
 But there is the issue that whether the word, "congruent", is appropriate to represent the concept.

Special-Student-7-Hypothesizer
 As is discussed above, supposing that the general meaning of 'congruent' is being same-shaped-and-same-sized, the mathematical "congruent" is against the general meaning or along the general meaning only by introducing the ambient space. Either way, I feel that that use of "congruent" somehow misleading.

Special-Student-7-Rebutter
 To summarize, 2 figures are said to be congruent iff 1 of them is the image of the other under an isometry, and the reason why isometry is used there is that isometry is along our typical concerns of whether 2 line segment lengths are equal and whether 2 angles are equal. Whether "congruent" is a desirable word is an issue, but an mirror image can be regarded to be same-shaped-and-same-sized by introducing the ambient space, although whether introducing such an extra space is wise is another issue.

7: Mirroring on a Plane or in a 3-Dimensional Space as a Rotation in the Ambient Space


Special-Student-7-Hypothesizer
 Although we do not really need the ambient space in order to define 'congruence', "One of the 2 figures can be moved in the ambient space to coincide with the other figure." seems to be a view of congruence, and let us see how any mirroring is a rotation in the ambient space.

A mirroring on a plane will be easy to imagine as a rotation in the ambient 3-dimensional space.

Let us suppose that there is a triangle, \(ABC\), on the \(0 \lt x\) half of the \(x-y\) plane, and \(ABC\) is mirrored with respect to the \(y\) axis, with the mirror image, \(A'B'C'\) on the \(x \lt 0\) half of the plane.

As one will be able to imagine, when \(ABC\) is rotated \(\pi\) around the \(y\)-axis in the ambient 3-dimensional \(x-y-z\) space, \(ABC\) coincides with \(A'B'C'\).


More specifically, when any point, \((x, y, z)\), is rotated \(\theta\) around the \(y\) axis, the image, \(x', y', z'\), is \(\begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} cos \theta & 0 & - sin \theta \\ 0 & 1 & 0 \\ sin \theta & 0 & cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}\), by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but \(x' = cos \theta x - sin \theta z, y' = y, z' = sin \theta x + cos \theta z\).

Special-Student-7-Rebutter
 \(y\) is not changed as we are rotating the point around the \(y\) axis.

Special-Student-7-Hypothesizer
 When \(z = 0\), which means that the point is on the \(x-y\) plane, and \(\theta = \pi\), \(x' = - x, y' = y, z' = 0\), which is indeed the mirroring.

Let us see a mirroring in a 3-dimensional space as a rotation in the ambient 4-dimensional space.

Let us suppose that there is a tetrahedron, \(ABCD\), in the \(0 \lt x\) half of the \(x-y-z\) space, and \(ABCD\) is mirrored with respect to the \(y-z\) plane, with the mirror image, \(A'B'C'D'\) in the \(x \lt 0\) half of the space.


Special-Student-7-Rebutter
 While the mirroring on the plane was done with respect to a line, the mirroring in the 3-dimensional space is done with respect to a plane.

Special-Student-7-Hypothesizer
 Yes. In fact, a mirror you use to look at your face is a plane, not a line, right?

Special-Student-7-Rebutter
 I have not used any line mirror before.

Special-Student-7-Hypothesizer
 More generally, a mirroring in a \(\mathbb{R}^n\) space is done with respect to a \(n - 1\)-dimensional hyperplane. In fact, the line on the plane is a \(1\)-dimensional hyperplane.

When \(ABCD\) is rotated \(\pi\) around the \(y-z\) plane in the ambient 4-dimensional \(x-y-z-w\) space, \(ABCD\) coincides with \(A'B'C'D'\).

Special-Student-7-Rebutter
 How can I imagine "rotated \(\pi\) around the \(y-z\) plane"?

Special-Student-7-Hypothesizer
 Let us think in this way: rotations around the \(y\)-axis in the \(x-y-z-w\) space are 3-dimensional rotations in the \(x-z-w\) space.

Special-Student-7-Rebutter
 Ah, as only the \(y\)-axis is fixed, it is free in the \(x-z-w\) space.

Special-Student-7-Hypothesizer
 Let us restrict the 3-dimensional rotations in the \(x-z-w\) space to only the rotations around the \(z\)-axis, which is what we are talking about.

Special-Student-7-Rebutter
 So, they are the rotations with the \(y\)-axis and the \(z\)-axis fixed.

Special-Student-7-Hypothesizer
 And such any rotation can be parameterized by a single angle, \(\theta\), which is the angle of the 3-dimensional rotation in the \(x-z-w\) space around the \(z\)-axis.

When any point, \((x, y, z, w)\), is rotated \(\theta\) around that way, the image, \(x', y', z', w'\), is \(\begin{pmatrix} x' \\ y' \\ z' \\ w' \end{pmatrix} = \begin{pmatrix} cos \theta & 0 & 0 & - sin \theta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ sin \theta & 0 & 0 & cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}\), by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but \(x' = cos \theta x - sin \theta w, y' = y, z' = z, w' = sin \theta x + cos \theta w\).

Special-Student-7-Rebutter
 It has to be that because it is the 3-dimensional rotation in the \(x-z-w\) space around the \(z\)-axis while the \(y\)-axis is already fixed.

\(y\) and \(z\) are not changed of course.

Special-Student-7-Hypothesizer
 When \(w = 0\), which means that the point is in the \(x-y-z\) space, and \(\theta = \pi\), \(x' = - x, y' = y, z' = z, w' = 0\), which is indeed the mirroring.

Special-Student-7-Rebutter
 So, while most people probably cannot easily imagine rotations in the ambient 4-dimensional space, any mirroring in any 3-dimensional space is indeed a rotation in the ambient 4-dimensional space.

Special-Student-7-Hypothesizer
 A way to imagine the rotation in the 4-dimensional space is to think of the projections of the rotation into the \(x-y-z, x-y-w, x-z-w, y-z-w\) spaces.





For example, the projection of \((x, y, z, w)\) into the \(x-y-w\) space is \((x, y, w)\), and \(B: (1, 0, 0, 0)\) and \(D: (1, 0, 1, 0)\) map to the same \((1, 0, 0)\); the projection of \((x, y, z, w)\) into the \(y-z-w\) space is \((y, z, w)\), and \(B: (1, 0, 0, 0), C: (4, 0, 0, 0), B': (-1, 0, 0, 0), C: (-4, 0, 0, 0)\) map to the same \((0, 0, 0)\).

Special-Student-7-Rebutter
 We have seen the projections, so, what?

Special-Student-7-Hypothesizer
 Well, an important thing is that when you look at a 3-dimensional space, you are aware that the space is not the whole space but a projection, which means that each point is not really a point but a set of points with 1 parameter.

Special-Student-7-Rebutter
 So, when I extend my finger to touch a point in the projection, I have to be aware that I may not really touch the point in the 4-dimensional space, because my finger is touching the \(w = 0\) one in the set of points while the point is at \(w = 2\) for example.

Special-Student-7-Hypothesizer
 "Should be touching the point but not really touching the point" may seem supernatural but is not: the point is just at another \(w\).

Special-Student-7-Rebutter
 I see, ..., but so what?

Special-Student-7-Hypothesizer
 Well, beyond that, it is a matter of your imagination; in an analogue, you cannot really see any 3-dimensional object, either: you see 2-dimensional projections and your brain imagines the shape of the 3-dimensional object.

Special-Student-7-Rebutter
 So, it should be the same with a 4-dimensional object, you mean.

Special-Student-7-Hypothesizer
 I am guessing that it is a matter of practices that you begin to imagine the 4-dimensional object.

References
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